Nordhaus–Gaddum-type inequality for the hyper-Wiener index of graphs when decomposing into three parts

Abstract

Let k≥2 be an integer, a k-decomposition(G1,G2,⋯,Gk) of a graph G is a partition of its edge set to form k spanning subgraphs G1,G2,…,Gk. The hyper-Wiener index WW is one of the recently conceived distance-based graph invariants (Randi 1993 [15]): WW=WW(G):=12W(G)+12W2(G), where W is the Wiener index (Wiener 1947 [18]) and W2 is the sum of squares of distance of all pairs of vertices in G. In this paper, we investigate the Nordhaus–Gaddum-type inequality of a 3-decomposition of Kn for the hyper-Wiener index: 7n2≤WW(G1)+WW(G2)+WW(G3)≤2n+24+n2+4(n−1). The corresponding extremal graphs are characterized.